A verified method for bounding clusters of zeros of analytic functions

In this paper, we propose a verified method for bounding clusters of zeros of analytic functions. Our method gives a disk that contains a cluster of m   zeros of an analytic function f(z)$f(z)$. Complex circular arithmetic is used to perform a validated computation of n  -degree Taylor polynomial p(z)$p(z)$ of f(z)$f(z)$. Some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p(z)$p(z)$. A validated computation of an upper bound for Taylor remainder series of f(z)$f(z)$ and a lower bound of p(z)$p(z)$ on a circle are performed. Based on these results, Rouché's theorem is used to verify that the disk contains the cluster of zeros of f(z)$f(z)$. This method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function. Numerical examples are presented to illustrate the efficiency of the proposed method.     

On the zeros of analytic functions belonging to Gevrey classes

For functionsf(z) ? 0, holomorphic in the unit disk u, infinitely differentiable in u, and belonging to a given Gevrey class on ?u, we establish sufficient conditions characterizing the sets K _f ^∞ = (z: ¦z¦ = 1,f ^(k) (z) = 0,k = 0, 1, 2, ... }. These conditions are close to the necessary condition due to L. Carleson and substantially more precise than the conditions given byA.-M. Chollet (see [1, 2]).     

Cyclicity of zeros of families of analytic functions

Let be a family of holomorphic functions in the unit disk , which are also holomorphic in a parameter . We express cyclicity (=generalized multiplicity) of a zero of at via some algebraic characteristics of the ideal generated by the Taylor coefficients of . As an example we estimate the cyclicity of the family of generalized exponential polynomials.     

Lower bounds for the zeros of analytic functions

Summary In the present work the problem of finding lower bounds for the zeros of an analytic function is reduced by a Hilbert space technique to the well-known problem of finding upper bounds for the zeros of a polynomial. Several lower bounds for all the zeros of analytic functions are thus found, which are always better than the well-known Carmichael-Mason inequality. Several numerical examples are also given and a comparison of our bounds with well-known bounds in literature and/or the exact solution is made.     

A modification of the classical quadrature method for locating zeros of analytic functions

The classical method for determination of a simple zeroa of an analytic functionf(z) inside a closed contourC by using the formulaa=(2i)^–1 _C [zf'(z)/f(z)]dz is reconsidered and modified. The modification consists in using the Cauchy theorem instead of the Cauchy formula and, further, evaluating numericallya as one of the zeros of an appropriate polynomial depending on the quadrature rule used. The case of circular contours with application of the trapezoidal rule is considered in detail and numerical results are presented.     

Counting functions of zeros of analytic functions in spaces with mixed norm

We study the behavior of counting functions of zeros of analytic in a disk functions in spaces with mixed norm, in particular, the Bergman-Dzhrbashyan spaces with standard weights. We obtain corollaries that strengthen the known results on zero sets of spaces with mixed norm.     

A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

Summary The argument principle is a natural and simple method to determine the number of zeros of an analytic functionf(z) in a bounded domainD. N, the number of zeros (counting multiplicities) off(z), is 1/2 times the change in Argf(z) asz moves along the closed contour D. Since the range of Argf(z) is (–, ] a critical point in the computational procedure is to assure that the discretization of D, {z _i ,i=1, ...,M}, is such that . Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of D lead to a completely reliable algorithm to computeN. This algorithm also treats in an elementary way the case when a zero is on or near the contour D. Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.Dedicated to Professor Ivo Babuka in commemoration of his sixtieth birthdayThis research is part of the doctoral dissertation of this author     

On small circles containing zeros and ones of analytic functions

Gol'dberg considered the class of functions with unequal positive numbers of zeros and ones inside the unit disc. The maximum modulus of zeros and ones in this class is bounded from below by a universal constant. This constant determines the limits of certain controller designs as well as covering regions of certain composites with schlicht functions. Considering lower bounds in a zero-free region of the extremal function the best known estimates of this constant are improved.     

Approximating the zeros of analytic functions by the exclusion algorithm

We give a practical version of the exclusion algorithm for localizing the zeros of an analytic function and in particular of a polynomial in a compact of . We extend the real exclusion algorithm to a Jordan curve and give a method which excludes discs without any zero. The result of this algorithm is a set of discs arbitrarily small which contains the zeros of the analytic function.     

On simple double zeros and badly conditioned zeros of analytic functions of variables

We give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros.

     

On zeros of functions of given proximate formal order analytic in a half-plane

We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ₊={z:Rez> and satisfy the condition

Application of the Cauchy theorem to the location of zeros of sectionally analytic functions

Summary A new method for the derivation of closed-form formulae for the zeros of sectionally analytic functions in the complex plane is proposed. This method is based on the Cauchy theorem in complex analysis (in a generalized form) and it does not require the solution of any boundary value problem along the discontinuity interval. As an application, the classical transcendental equationw=tanh (p w+q), appearing in a physical problem, is solved in closed form. Numerical results are also presented.

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Résumé Une nouvelle méthode pour la dérivation de formules analytiques pour les zéros des fonctions analytiques par morceaux dans le plan complexe est proposée. Cette méthode se base sur le théorème de Cauchy dans l'analyse complexe (sous une forme généralisée) et elle n'a besoin de la solution d'aucun problème à valeurs aux limites le long de l'intervalle de discontinuité. Comme une application, l'équation transcendentale classiquew=tanh (p w+q), apparaissante dans un problème physique, est résolue analytiquement. Des résultats numériques sont présentées aussi.

     

A simple quadrature-type method for the computation of real zeros of analytic functions in finite intervals

A new direct approximate method for the computation of zeros of analytic functions is proposed. For such a function possessing one real zero in a finite part of the real axis, this method permits the determination of this zero with a satisfactory accuracy by using a quite elementary algorithm. The present method is based on the Gauss- and Lobatto-Chebyshev quadrature rules for Cauchy type principal value integrals and is always convergent. The simplicity, accuracy and unrestricted convergence of the proposed method make it competitive to the analogous classical elementary methods. Numerical results are also presented.     

Description of sequences of zeros of one class of functions analytic in a half-plane

We describe sequences of zeros of functions ƒ ≠ 0 that are analytic in the right half-plane and satisfy the condition |ƒ(z)| ≤ 0(1) exp (σ| z |η(| z |)), 0 ≤ <+ ∞, Re z > 0, where η: [0; + ∞) → (- ∞; + ∞) is a function of bounded variation. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1169–1176, September, 1998.     

Specialization of Appell's functions to univariate hypergeometric functions

Univariate specializations of Appell's hypergeometric functions F₁, F₂, F₃, F₄ satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2 and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.